Most of us would consider ourselves to be a separate individual, and probably only a single individual. But, of course, you are made up of large numbers of separate cells. In constrast, many organisms such as amoeboids and ciliates are separate individuals with only a single cell. Bridging the gap between these two are colonial unicellular organisms such as many choanoflagellates. Attempting to establish a clear line between what should be consider a colony of unicellular organisms and a single multicellular organism is a futile task as classically demonstrated by the example of sponges, made up of multiple distinct cell-types that are derived from a single ancestral zygote but retain the ability to function independently of the remainder of the sponge*. Amoeboid sponge cells may actually detach themselves from the base of the colony/individual and move off on short independent 'exploratory expeditions' before returning to the remainder of the sponge and re-integrating themselves**. The ability of sponges to re-assemble themselves after being divided into separate cells is also well-known (Galtsoff, 1923; if disassociated cells from two or more separate sponges are mixed together, they will re-assemble themselves back into separate sponges).
*Apparently, this is known as the "sorites paradox". 'Sorites' is Greek for 'something that is piled up', and the name relates to the question of how many grains of sand it takes for a cluster of grains to be considered a pile of sand. Hence, if I understand this correctly, a sorites problem applies to situations where you have two clearly distinct end-states, but no clear dividing line between them. Just to confuse matters, Sorites is also the name of a genus of foraminiferans.
**One small detail that you may not realise - as a result of this movement of amoeboid cells, the entire sponge is capable of very slow movement, a few millimetres over the course of a day.
Even if one tries to define the cell as individual, nature still has tricks up her sleeve. Multinucleate cells, for instance, can go through nuclear division without necessarily going through cell division, or go through cell division without going through nuclear division (anyone remember Trichosphaerium?) Hyphal or coenocytic organisms may be decidedly cavalier about whether or not they divide into separate cells. Even if the hyphae are transected by cell walls as in fungi, the walls may still have open pores that allow the free movement of nuclei between 'cells'. Plants also have connecting pores through cell walls called plasmodesmata; while plasmodesmata do not allow nuclear movement between cells, they do connect the cytoplasm of separate cells. Depending on how you chose to define a 'cell', it might be possible to regard an entire tree as a unicellular organism.
Things become even more complicated when you consider the possibility of separate 'individuals' within a single cell. For instance, there are intracellular parasites such as some red algae where free parasite nuclei are injected directly into the host cytoplasm without a cell membrane separating host and parasite. Fungi include some of the best examples of 'super-individual' cells, because of the ability of members of the Ascomycota and Basidiomycota (the two groups that include most macroscopic fungi) to form what is called a 'dikaryon'. As described by the Watcher in the post linked to at the top of this post, fertilisation in fungi occurs when two compatible lots of hyphae meet and fuse to exchange nuclei. However, unlike fertilisation in animals such as ourselves where the nuclei fuse right away to form zygotes, in fungi the two parent nuclei remain separate within the cytoplasm (fungi nuclei are normally haploid). Cell growth and division in the hypha continues as normal, with division of each of the separate nuclei maintaining the dikaryotic state in each of the new cells. It is these dikaryotic hyphae that make up the fruiting body, and fusion of the nuclei to form diploid zygotes does not happen until the actual point of spore production. So every mushroom you see represents the fusion of two separate parents, and contains (at least) two separate lines of nuclei. Is a mushroom one individual or two?
Galtsoff, P. S. 1923. The amoeboid movement of dissociated sponge cells. Biological Bulletin 45 (3): 153-161.